Nonlinear spectral model for rotating sheared turbulence

Abstract

We propose a statistical model for homogeneous turbulence undergoing distortions, which improves and extends the MCS model by Mons, Cambon & Sagaut (J. Fluid Mech., vol. 788, 2016, 147–182). The spectral tensor of two-point second-order velocity correlations is predicted in the presence of arbitrary mean-velocity gradients and in a rotating frame. For this, we numerically solve coupled equations for the angle-dependent energy spectrum ${\mathcal{E}}(\boldsymbol{k},t)$ that includes directional anisotropy, and for the deviatoric pseudo-scalar $Z(\boldsymbol{k},t)$, that underlies polarization anisotropy ($\boldsymbol{k}$ is the wavevector, $t$ the time). These equations include two parts: (i) exact linear terms representing the viscous spectral linear theory (SLT) when considered alone; (ii) generalized transfer terms mediated by two-point third-order correlations. In contrast with MCS, our model retains the complete angular dependence of the linear terms, whereas the nonlinear transfer terms are closed by a reduced anisotropic eddy damped quasi-normal Markovian (EDQNM) technique similar to MCS, based on truncated angular harmonics expansions. And in contrast with most spectral approaches based on characteristic methods to represent mean-velocity gradient terms, we use high-order finite-difference schemes (FDSs). The resulting model is applied to homogeneous rotating turbulent shear flow with several Coriolis parameters and constant mean shear rate. First, we assess the validity of the model in the linear limit. We observe satisfactory agreement with existing numerical SLT results and with theoretical results for flows without rotation. Second, fully nonlinear results are obtained, which compare well to existing direct numerical simulation (DNS) results. In both regimes, the new model improves significantly the MCS model predictions. However, in the non-rotating shear case, the expected exponential growth of turbulent kinetic energy is found only with a hybrid model for nonlinear terms combining the anisotropic EDQNM closure and Weinstock’s return-to-isotropy model.

Modal Projection for Quasi-Homogeneous Anisotropic Turbulence

“…The orthonormal frame is recovered by Ying et al ( 2019) [6] for the gradient operator in k-space, passing from Cartesian coordinates to polar-spherical ones, with…”

“…ZCG is a statistical spectral model developed for homogeneous anisotropic turbulence by the author Y. Zhu, C. Cambon, and our collaborator F. S. Godeferd. For all relevant formulas, numerical details, and simulation results presented in this section, part of them are published in Zhu et al (2019) [6], and the other unpublished part is in the Ph.D. thesis of the first author [14].…”

“…Neutral cases are found for R = 0 (no background rotation) and R = −1 (zero absolute vorticity). Accordingly, four typical rotating shear flows (with various values of R) are numerically simulated in [6] by ZCG, namely using ( 22) and ( 23): R = −5 corresponds to a stabilizing, anticyclonic case; R = −1 for a neutral case with zero absolute vorticity, as encountered in the central region of a rotating channel; R = −1/2 for a maximum destabilization, anticyclonic case, as in the pressure side of a rotating channel; and R = 0 with no rotation. The mean shear and system rotation are chosen as A ij = Sδ i1 δ j2 and Ω = Ωδ i3 , respectively, to vanish the term of Ω E in Equation (22).…”

“…Applications to statistical moments, such as spectral tensors related to two-point second-order vector correlations, are addressed in Section 3. Rotating shear turbulence is investigated in Section 4 by extending the recent model by Zhu et al (2019) [6]. In Section 5, a modal projection of poloidal/toroidal type, more general than the one related to Herring's one, is investigated for a vector field, such as a smooth divergence-free velocity field.…”

CFD for turbulence: from fundamentals to geophysics and astrophysics

“…In a recent paper, the convection operator by the mean flow, with spaceuniform gradients, is treated by a new scheme, that do not need distortion of the grid and related interpolation (from X − K discretized variables to "Eulerian" ones x − k, see Figure 2). In addition to application to a spectral statistical model, by Zhu et al [24], an alternative for DNS (versus Rogallo or Lesur) is in progress.…”